- Email: [email protected]
Monitoring multivariate profile data in plastic parts manufacturing industries: An intelligently data processing
Accepted Manuscript
Monitoring Multivariate Profile Data in Plastic Parts Manufacturing Industries: An Intelligently Data Processing Karim Atashgar , Omid Asghari Zargarabadi PII: DOI: Reference:
S2452-414X(16)30094-2 10.1016/j.jii.2017.06.003 JII 41
To appear in:
Journal of Industrial Information Integration
Received date: Revised date: Accepted date:
3 November 2016 23 February 2017 18 June 2017
Please cite this article as: Karim Atashgar , Omid Asghari Zargarabadi , Monitoring Multivariate Profile Data in Plastic Parts Manufacturing Industries: An Intelligently Data Processing, Journal of Industrial Information Integration (2017), doi: 10.1016/j.jii.2017.06.003
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ACCEPTED MANUSCRIPT Monitoring Multivariate Profile Data in Plastic Parts Manufacturing Industries : An Intelligently Data Processing
Karim Atashgar1 Malek Ashtar University of Technology, Department of Industrial Engineering, Tehran 15875-1774, Iran
Omid Asghari Zargarabadi
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Email: [email protected]
Malek Ashtar University of Technology, Department of Industrial Engineering, Tehran 15875-1774, Iran
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Email: [email protected]
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Abstract: In some real cases, more complicated models such as multivariate profile, rather than simple linear profile is addressed to model the performance of the process because of the existence of correlation between the response variables. In this case, if one ignores the correlation structure of response variables by assuming separate profiles then misleading results are expected. The complicated approach, in practice, is capable of leading practitioners to analyze parameters of the performance of the process effectively. This advanced approach also helps industrial information engineers to provide the information and communication technology (ICT) effectively for real industrial cases. Extrusion is a process that is extensively used in different plastic parts manufacturing industries. Technical analysis and our statistical analysis (in this research) indicate that there are correlations between controllable technical variables of an extrusion process. In this study, two important correlated response variables, namely, flow rate per unit length and mass per unit area characteristics are considered. Monitoring jointly these two important quality factors with multivariate profile supports integration purposes of industrial information effectively. In this paper Wilks’ lambda statistic, is used for analysis of the performance of the proposed multivariate profile model in phase-I of statistical control. The performance of the proposed multivariate profile model is investigated using test power term through several different numerical cases. The proposed approach is capable of facilitating information technology (IT) activities of the studied manufacturing industry. Keywords: Multivariate profile, Statistical process control, Extrusion, Phase-I of control
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1 .Introduction In statistical process control, quality of a process or a product is monitored traditionally by a random variable or by a vector involving several correlated quality characteristics. However, really, in many practical cases, the quality of a process or a product should be described by a relationship between one or more response(s) variable(s) and one or more independent variables. This advanced regression approach is referred to as the profile monitoring. The profile approach provides more realistic capability of monitoring product/ process characteristics comparing to conventional methods. The new approach can be attended in the second level of the discipline structure of industrial information integration engineering (IIIE) introduced by Xu [1]. 1
Corresponding author. Tel/Fax: +982122970309 E-mail address: [email protected] (K. Atashgar)
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Chen [2] also emphasized on new approaches for manufacturing industries by a comprehensive literature review of industrial information integration. In recent years different researchers have reported several studies approaching profile. Several authors such as Kang and Albin [3], Kim et al. [4], Mahmoud and Woodall [5], Yeh et al. [6], Vaghefi et al. [7], Kazamzadeh et al. [8],Amiri et al. [9] ,Colosimo et al. [10,11], Wang and Tsung [12], Zou et al. [13], Mestek et al. [14], Gupta et al. [15], Zhang et al. [16] , Saghaei et al. [17], Moguerza et al. [18], Qui et al. [19], Jin and Shi [20], Ding et al. [21], Shang et al. [22], Jensen et al. [23], Atashgar et al. [24], Khedmati and Niaki [25] have contributed to linear and nonlinear univariate profile monitoring. Literature addresses that Amiri et al. [26], Noorossana et al. [27], Zou et al. [28] and Soleimani and Noorossan [29], focused on monitoring a multivariate profile of phase-II. In this arena Noorossana et al. [30], Zhang et al. [31], and Paynabar et al. [32] have contributed to phase-I of multivariate profile monitoring. Noorossana et al. [30] considered two correlated regression equations in order to generate simulated data. In this consideration they generated simulated data based on assumed marginal normal distribution of each response variable. They assumed that there is an assumed covariance matrix and the distribution of the correlated response variables follows a bivariate normal because of simulated data of each response variable follows a marginal normal distribution. The approach used by Noorossana et al. [30] may leads to misleading signals. The bivariate normal test of the joint response variables of the example has not been also reported by Noorossana et al. [30]. The Noorossana et al. [30] report provides an evaluation analysis without regarding inverse shift directions. In Table 1 the literature is summarized. As shown in Table 1 univariate type of profile is focused on by most authors. Extruder process is used extensively in different plastic parts manufacturing industries. An effective method to monitor this process considerably helps practitioners to control technical factors of the process. Extruder is a machine which takes raw materials mostly consisting of rubber or plastic polymers and then produces products in different shapes. In this process two important quality specifications i.e. flow rate per unit length and mass per unit area should be controlled by quality engineers. Technical analysis indicates that flow rate per unit length and mass per unit area are affected by mold temperatures (see Rauwendaal [33].The technical report of Rauwendaal [33] indicates that the two specifications are also correlated. Table 1: Profile resources of literature
Phase-I
CE Phase-II
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No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 16 17 18 19 20 21
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Phase type
Mahmoud &Woodall [5] Ding Y et al [21] Gupta et al [15] Moguerza et al [18] Eghbali et al [34] Kazemzadeh et al [8] Mahmoud [35] Colosimo et al. [10] Noorossana et al [30] Amiri et al [9] Amiri et al. [36] Yeh & Zerehsaz [37] Amiri et al [38] Zhang et al. [31] Atashgar et al [24] Paynabar et al. [32] Jin & Shi [20] Kang & Albin [3] Kim et al [4] Saghaei et al. [17] Soleimani et al. [39] Vaghefi et al. [7]
Profile type Univariate Multivariate ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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Jensen et al. [23] Colosimo et al [11]
● ● ● ● ●
Ebadi & Amiri [40] Eyvazian et al. [41] Noorossana & Soleimani [42] Soleimani & Noorossana [29] Khedmati & Niaki[ [25]
● ● ●
Amiri et al. [26]
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This study focuses on monitoring flow rate per unit length and mass per unit technical characteristics using profile approach. This research analyzes the performance of the proposed multivariate profile for phase-I. The statistical analysis of this study confirms the existence of correlation between flow rate per unit length and mass per unit claimed in the technical analysis of Rauwendaal [33]. In this study the mold temperature of the extrusion process is allowed to play as an independent variable for the multivariate profile. This approach provides the required capability of a smart factory for plastic parts manufacturing industries basically. Kim and Parek [43] analyzed Industry 4.for small batch production. The investigation conveys the idea of the integration production management systems (i.e. technical characteristics monitoring and etc.) with developing new technologies. The profile approach leads to accuracy of inventory data as emphasized by Alyahya et al. [44]. Branger and Pang [45] discussed about the domain of the vision of future homes including automated home, sustainable home, healthy home, and manufacturing home. The profile monitoring can be used in all sections of the future homes.
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The rest of this paper is organized as follows: Section 2 introduces the studied extrusion process. Multivariate profile monitoring is discussed in section 3. Our proposed model to control the performance of an extruder and statistical analyzes are explicated in section 4. The evaluation report and our concluding remarks are provided in sections 5 and 6, respectively.
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2. Briefing the extruder process An extruder machine is used to produce different plastic/ rubber parts. In this process a compound is passed through a mold under a high pressure, so that the material is allowed to form in different cross sections. In the studied extruder process, components are produced continuously and the input is polymer material (PE). In this process, the output is produced in the shape of thin sheet continuously. In this process uniform mass distribution over the sheet and flow rate per unit length of the sheet are two crucial technical characteristics. Figure 1 addresses the main components of the extruder machine. Insert Figure 1 about here
In this process mold temperature is also a crucial specification. When temperature of a mold decreases or increases, viscosity of material is allowed to change. Rauwendaal [33] explained that flow rate per unit length and mass per unit area characteristics are two correlated variables. As addressed by Rauwendaal [33] dynamics of the output flow of an extruder is influenced directly by the mold temperature. The flow dynamics affects both flow rate per unit length and mass per unit area factors. In this paper the correlation of these two variables is also addressed statistically. In this paper to control the performance of an extruder machine, a multivariate profile model is proposed. In addition the performance analysis of this process is provided for phase-I of statistical quality control. 3. Multivariate profiles
ACCEPTED MANUSCRIPT Generally, linear profiles can be classified into two branches; 1) single response variable, and 2) multi correlated response variables (multivariate profile). In a multivariate profile there are more than one response variable. In this case each response is defined as a function of an independent variable(s), while the response variables mutually affect each other. Assume that there are m sample sets for an independent variable x, each of which includes n observations, and there are p response variables for each independent variable. In this case there are n observations for the kth sample as follows. (x i , yi 1k , yi 2 k ,...., y ipk )
(1)
Yk XBk Ek y 12 k y 22 k y n 2k
where
y 1 pk 1 x 1 1 x y 2 pk 01k 2 11k y npk 1 x n
Yk (y1k , y 2 k ,..., y nk )T is
02 k 12 k
0 pk 1 pk
11k 21k n 1k
12 k 22 k
n 2k
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y 11k y 21k y n 1k
(2)
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where x i indicates the ith sample value from the jth sampling for p response variables of y i 1k , y i 2k ,..., y ipk . For each sample, the model addresses a relationship between each response variable and an independent variable as follows. 1 pk 2 pk npk
a n p response variables matrix for the k sample. Each row of th
(2)
Yk
corresponds to one value of independent variable ( x i ) and includes p values for response variables. Let
n 2 dimension, where x includes the values of the denotes a n p error matrix for the kth sample. The rows of
X [1 x] be given as an independent matrix with a
independent variable. Matrix
Ek (ε1k , ε 2 k ,...ε nk )
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Ek follow independently an identical multivariate normal distribution with 1 p zero mean vector and p p covariance matrix Σ . In another word, elements of each row of Ek are dependent; however, each row is
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addressed independently. In Equation 2, coefficient matrix
Bk (β0 k , β1k )T
denotes a 2 p dimension. This
ˆ )T (Y XB ˆ ) as the following matrix is estimated from least squares method which minimizes (Yk XB k k k
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equation.
y .1k ˆ1 jk x Sxy (1) Sxx
ˆ0 pk ˆ1 pk
y . pk ˆ1 jk x Sxy ( p ) Sxx
(3)
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where
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ˆ ˆ (XT X)-1 XT Y (βˆ , βˆ ) T 01k B k k 0k 1k ˆ 11k
n n 1 n 1 n x x i ، S xy ( j ) (x i x ) y ijk ، y . jk y ijk ، S xx (x i x )2 n i 1 n i 1 i 1 i 1
(4)
3.1. Wilks’ lambda statistic approach In this research Wilk’s statistic is used to analyze the proposed model. The statistic is developed based on F-statistic proposed by Mahmoud and Woodall [5]. Mahmoud and Woodall [5] proposed the use of indicator or artificial variables for a multiple regression. The method is developed for a multivariate regression. Consider m samples each of which is included of n observations as follows: k =1,2,…,m; i = 1,2,…,n. (x ik , y i 1k , y i 2 k ,..., y ipk ) (5)
ACCEPTED MANUSCRIPT The aim is to examine the equality of regressions of different samples. For this purpose, all m samples are pooled and a sample set including mn members is constructed. In this stage it is possible to define m-1 indicator variables as follows. {
.
(6)
In this case it is allowed to define the following test statistic of null hypothesis H 0 . ˆ X Y YY B f f . ˆ YY Br Xr Y where
Y (y1 , y 2 ,..., y mn ) T
(7)
is a mn p matrix of response variables for the pooled variables. In Equation 7,
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Xf denotes a mn 2m matrix of independent variables of the full model, and Bˆ f indicates a 2m p matrix including estimated coefficients of the full regression model. However, in this equation Xr with mn 2
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dimension and Bˆ r with 2 p dimension address the reduced model. Under a null hypothesis, this statistic follows Wilks’ lambda distribution with p, 2(m-1) and m (n-2) degrees of freedom. The null hypothesis is rejected when the test statistic is less than , p ,2( m 1),m ( n 2) . In this research, for the equality test of a covariance matrix, the method proposed by Lee et al. [46] is utilized. For this purpose, unbiased estimation of covariance matrix for the kth sample is denoted as follows.
Yk Yk Bˆ k X Yk . (8) n 2 where Yk is a n p matrix of response variables, X [1 x] is a n 2 matrix of independent variable, and Bˆ k is Sk
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the estimation of least squares. In order to generate the test statistic, M is calculated as the below equation 9. ( n 2)/2 ( n 2)/2 ( n 2)/2 S1 S2 Sm ( S 1 S 2 S m )( n 2)/2 M m( n 2)/2 m( n 2)/2 . (9) S S S pl is
m
S pl
k 1 m
the pooled value of the covariance matrix which is determined as follows. (n 2)S k
(n 2) k 1
m k 1
Sk
m
(10)
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where
pl
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pl
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When M value is allowed to be close to 1, it is addressed that the equality hypothesis is not rejected, whereas in the case that the value is close to zero value, it is allowed to reject the hypothesis. The reader for exact critical value of -2lnM statistic is referred to Lee et al. [46]. Noorossana et al. [47] addressed that the developed Fstatistic method called Wilk’s statistic is superior compared to other method proposed in literature of multivariate cases. In this paper, this method is used to monitor the proposed multivariate profile of the extruder process. 4. Phase I extruder profiles monitoring using Wilks’ lambda statistic method As mentioned before flow rate per unit length and mass per unit area are considered as two important response variables in the process of extruder, while the mold temperature characteristic is an explanatory variable in this process. In this research a digital displayer and a digital scale are employed to record data of the flow rate per unit length and the mass per unit area respectively. Response variable values are provided for 11 different levels (220 to 240 with distances 2). Since the temperature factor differs point by point of a mold; hence, in this study the averaged measure is utilized. Considering aforementioned information and 3 repetitive sampling in each level of independent variable, in this case n (the size of a sample) is addressed by Noorossana et al. [47]. The sampling is done 30 times i.e. m equals to 30, and overall, there are 30 samples each of which
ACCEPTED MANUSCRIPT includes 33 observations leading to 990 random experiments. To calculate Wilk’s lambda statistic , Bˆ f and
Bˆ r are obtained from Equation 11. Bˆ r (XT X)-1 XT Yr , Bˆ (XT X)-1 XT Y f
(11)
f
1.8187 1.5074 ˆ Calculations lead to Bˆ r , however Bf is a 2 60 matrix which is provided in the appendix. 0.0262 0.0164 In this study Wilk’s statistics equals to 0.86675 . Since , p ,2( m 1),m ( n 2) 0.025320,2,58,930 0.85736 0.86675 is addressed, the null hypothesis is not rejected. It means that the process is in-control in phase-I.
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Table2. Covariance matrix of each profile
S2 S3 S4 S5 S6
0.8528
0.0069
0.0037
1.5802
0.8522
0.0069
0.0037
1.5726
0.8493
0.0068
0.0037
1.4269
0.7690
0.0062
0.0033
1.4091
0.7596
0.0061
0.0033
1.4490
0.7802
0.0063
0.0034
S7 S8 S9 S10 S11 S12
1.5855
0.8539
0.0069
0.0037
1.6030
0.8629
0.0070
0.0037
1.6077
0.8648
0.0070
0.0038
1.6660
0.8959
0.0072
0.0039
1.6227
0.8731
0.0070
0.0038
1.6203
0.8727
0.0070
0.0038
S13 S14 S15 S16 S17 S18
1.6117
0.8672
-0.0070
0.0038
1.6234
0.8737
-0.0071
0.0038
1.6172
0.8716
-0.0070
0.0038
1.7075
0.9205
-0.0074
0.0040
S19 S20 S21 S22
1.6285
0.8780
-0.0071
0.0038
1.6903
0.9114
-0.0073
0.0040
1.6038
0.8650
-0.0070
0.0038
1.6318
0.8800
-0.0071
0.0038
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S1
1.5811
1.6547
0.8914
-0.0072
0.0039
1.7291
0.9321
-0.0075
0.0040
S23 S24
1.5939
0.8606
-0.0069
0.0037
1.4195
0.7650
-0.0062
0.0033
S25 S26 S27 S28 S29 S30
1.4514
0.7824
0.0063
-0.0034
1.4263
0.7679
0.0062
-0.0033
1.6219
0.8757
0.0070
-0.0038
1.6037
0.8649
0.0070
-0.0038
1.5939
0.8606
0.0069
-0.0037
1.4387
0.7754
0.0062
-0.0034
M
As discussed before to test the covariance matrix equality, the method proposed by Lee et al. [46] is used. The results are listed in Table1 for 30 samples of this study. Each S ( ) shown in Table 2 addresses the variance-covariance matrix of each m sample. The calculation leads to the following matrix for S pl .
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1.57905 0.846245 S pl 0.00686 0.00370 In this step using Equation 9, M is derived to be M 0.985185185 . Since M statistic is very close to 1, the covariance equality hypothesis is confirmed.
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5. Proposed Model evaluation In this evaluation the covariance matrix of the multivariate regression is calculated using Equation 12 (where N and r respectively denote total number of samples and total number of estimated parameters). This matrix allows us to produce simulated data for the studied process. In this paper the following cases are considered for analyzing the proposed profiles: 1) Synchronous and asynchronous shift cases. 2) Shifts with the same and inverse directions corresponding to parameters of the multivariate profile. To address the normality of the correlated response variables Johnson and Wichern [48] method is used. To evaluate the performance of the proposed multivariate profile each combination of shift size is simulated for 10000 iterations. In this evaluation, equations 12 and 13 obtained from phase-I of the extruder process are utilized. In these equations n=33, m=30 and ( i 1k , i 2k . i 3k , i 4k ) follow multivariate normal distribution with zero mean vector and covariance of . The estimation of is estimated from Equation 12.
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^
^
^
ε ε (Y-Z )' (Y-Z ) 0.5599 0.1361 3 10 0.1361 0.199 3 N-r-1 N-r-1 ^
(12)
Equation13 addresses the fitted multivariate profile model based on data for phase-I of the extruder process.
Y i 1k 1.8187 0.0262 x i i 1k
(13)
i 1, 2,, n , k 1, 2, , m
Y i 2k 1.5074 0.0164 x i i 2k
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Figure 2 shows the correlation of profile versus profile using simulated data obtained from 10000 iterations. As shown in Figure 2 the correlation between the profiles is negative. It should be stated that the variance of the first profile is larger compared to the second one. In this paper to evaluate the proposed model the following shifts are examined: 1) Shift in β 0 with different amount values, for various samples and in different directions (i.e. an intercept parameter decreases while another one is allowed to increase). 2) Shift in β1 with different amount values, for various samples and in different directions (i.e. a slope increases while another one decreases).
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3) Different shift values in for different samples and in different directions (i.e. one of the variances increases while another one is allowed to decrease). In this analysis type-I error probability for Wilks’ lambda statistic and M statistic of Lee et al. [46] is assumed to be α 2 1 1 α 0.02532 so that the total type-I error becomes 0.05. These shifts are ^
simultaneously induced into profiles 1 and 2. The intercept shift is in (1/ n ) (X 2 / s xx ) unit, the shift of
M
^
^
slope is in / s xx unit, and the shift values of covariance are in a factor of .
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Insert Figure 2 about here
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5.1. Normality test of response variables If several variables follow a joint distribution such as a multivariable normal distribution, the marginal distribution of each of the variables follows single normal distribution; nevertheless, the inverse is not addressed necessarily. In other words, the joint distribution of the variables does not necessarily follow a normal multivariable distribution in the case that each distribution of variables is addressed by a normal distribution (for more details, see Johnson and Wichern [48]). Hence if each row of Y follows for example a bivariate 2 normal distribution, it is expected that χ p, 0.0201 with error value 0.01 , and the number of dependent variables p=2. In this case it is expected that 99% of a desired bivariate distribution points are located inside the ellipse represented by Equation 14. (Y-Y)' S-1 (Y-Y) χ 22,0.01 (14) Now for the case of this research the following equation is obtained: /
y 1 4.20 0.0279 0.017 y 1 4.20 2 χ 2,0.01 0.0201 y 2 2.26 0.017 0.011 y 2 2.26 To check normality of the studied multivariate profile, Equation 14 is coded. The results revealed that normality hypothesis is acceptable. 5.2. Test power for intercept, slop, and covariance shifts
ACCEPTED MANUSCRIPT Figures 3 to 8 and Table 2 provide out-of-control signal probability for different shifts from 10000 iterations. The results indicate a high capability for the proposed model when the intercept parameter is affected by different shift sizes. As shown in Table 2 the best performance is addressed when a change occurs for the middle sample (that is to say, the best performance is obtained when shifts occur in sample 15). When changes get far from the middle sample, the performance addresses degradation, so that there is symmetry between performance degradation in the first and the last samples. It should be stated that test power index is allowed to decrease when a same direction of shifts are experienced by the process. It is interesting that even in this case, the best test power value is addressed by the middle sample.
shift in first profile
shift in first profile
^
sustained shifts in β 0 for sample 5
shift in second profile
shift in second profile
with same directions
sustained shifts in β 0 for sample 5 with same directions
0.01 0.02 0.03 0.03 0.04 0.03 0.05
0.01 0.063 0.126 0.312 0.615 0.885
0.02 0.122 0.273 0.53 0.838 0.959
0.03 0.299 0.527 0.786 0.935 0.999
0.04 0.616 0.799 0.947 0.995 1
^
sustained shifts in β 0 for sample 15 with same directions
sustained shifts in β 0 for sample 15 with same directions
0.01 0.02 0.03 0.04 0.05
0.01 0.076 0.282 0.688 0.948 0.997
0.02 0.304 0.607 0.903 0.996 1
0.03 0.69 0.906 0.996 1 1
^
0.04 0.968 0.998 1 1 1
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for sample 25 with same directions
0.01 0.058 0.115 0.296 0.653 0.887
0.02 0.139 0.279 0.524 0.83 0.953
0.03 0.303 0.539 0.75 0.947 0.996
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sustained shifts in β 0
0.01 0.02 0.03 0.04 0.05
0.04 0.611 0.809 0.939 0.994 0.998
for sample 5 with inverse directions
0.01 0.02 0.03 0.03 0.04 0.03 0.05
0.01 0.04 3 0.07 3 0.15 1 0.35 4 0.66 2
0.02 0.07 4 0.09 6 0.16 2 0.37 1 0.61 4
0.03 0.15 2 0.14 8 0.24 6 0.42 4 0.68 3
0.04 0.35 8 0.34 2 0.44 3 0.58 0.75 5
0.05 0.66 8 0.63 9 0.67 5 0.75 6 0.86
sustained shifts in β 0 for sample 15 with opposite directions
0.05 1 1 1 1 1
sustained shifts in β 0 and sample 25 with same directions ^
^
sustained shifts in β 0
^
M
^
0.05 0.879 0.965 0.994 0.998 1
^
sustained shifts in β 0 for sample 5 with inverse directions
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Table3. Intercept shift values in different locations and the same and inverse directions
0.05 0.885 0.964 0.994 0.999 1
^
sustained shifts in β 0 for sample 15 with inverse directions
0.01 0.02 0.03 0.04 0.05
^
0.01 0.04 1 0.11 3 0.35 5 0.73 9 0.97 5
0.02 0.1 0.19 0.37 7 0.74 7 0.95 7
0.03 0.36 0.37 6 0.55 8 0.83 1 0.97 8
0.04 0.78 4 0.76 2 0.83 9 0.94 6 0.99
0.05 0.97 0.96 3 0.97 3 0.98 8 1
sustained shifts in β 0 and sample 25 with inverse directions ^
sustained shifts in β 0 for sample 25 with inverse directions
0.01 0.02 0.03 0.04 0.05
0.01 0.04 6 0.05 3 0.14 8 0.35 9 0.71
0.02 0.07 2 0.08 8 0.17 5 0.33 0.62
0.03 0.15 8 0.16 3 0.25 9 0.42 9 0.66
0.04 0.37 7 0.32 4 0.42 5 0.56 0.76
0.05 0.66 8 0.65 0.70 1 0.75 9 0.88
Insert figures 3-8 about here ^
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The following figures and Table 3 provide out-of-control signals when β1 parameter experiences ^
^
different sustain shifts i.e. β1 / S xx . As shown in Table 4 the best performance is addressed in the middle sample i.e. the 15th sample. When changes get far from the middle sample, the performance is allowed to degrade. In this case there is symmetry between performance degradation. It should be stated that the test power term decreases when slope shifts are not addressed by a same direction.
ACCEPTED MANUSCRIPT Table4. Slope shift values in different locations and in the same and inverse directions shift in first ^ profile ^ sustained shifts in β1 for sample 5 shift in sustained shifts in β1 for sample 5 second profile with same directions with inverse directions
0.01
0.02
0.03
0.04
0.05
0.01
0.05
0.11
0.32
0.62
0.87
sustained shifts in β1
0.02
0.13
0.25
0.52
0.81
0.96
for sample 5 with same directions
0.03
0.30
0.52
0.76
0.93
0.99
0.04
0.60
0.81
0.93
0.99
0.99
0.05
0.88
0.95
0.99
1
1
^
^
for sample 15 with same directions
0.01
0.02
0.03
0.04
0.05
0.11
0.26
0.68
0.95
0.99
0.02
0.29
0.60
0.90
0.99
0.99
0.03
0.67
0.91
0.98
1
1
0.04
0.93
0.99
1
1
1
0.05
0.99
1
1
1
1
^
0.01
0.02
0.03
0.04
0.01
0.05
0.11
0.29
0.6
sustained shifts in β1
0.02
0.11
0.25
0.50
0.79
for sample 25 with same directions
0.03
0.30
0.52
0.75
0.95
0.04
0.62
0.80
0.94
0.05
0.87
0.95
0.99
0.05
0.01
0.03
0.06
0.13
0.35
0.65
sustained shifts in β1
0.02
0.07
0.08
0.15
0.31
0.62
for sample 5 with inverse directions
0.03
0.12
0.15
0.23
0.43
0.66
0.04
0.34
0.35
0.39
0.56
0.75
0.05
0.67
0.63
0.64
0.75
0.86
^
^
sustained shifts in β1
for sample 15 with inverse directions
0.01
0.02
0.03
0.04
0.05
0.01
0.05
0.11
0.37
0.75
0.97
0.02
0.09
0.19
0.38
0.73
0.97
0.03
0.32
0.39
0.55
0.83
0.97
0.04
0.74
0.73
0.82
0.93
0.99
0.05
0.96
0.96
0.97
0.98
0.99
^
0.05
0.01
0.02
0.03
0.04
0.05
0.87
0.01
0.03
0.06
0.13
0.37
0.66
^
0.95
sustained shifts in β1
0.02
0.06
0.07
0.18
0.36
0.62
0.99
for sample 25 with inverse directions
0.03
0.14
0.14
0.21
0.38
0.66
0.04
0.35
0.34
0.42
0.50
0.73
0.05
0.66
0.62
0.65
0.75
0.86
0.99
0.99
0.99
1
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^
0.04
sustained shifts in β1 and sample 25 with inverse directions
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0.03
sustained shifts in β1 and sample 15 with inverse directions
0.01
sustained shifts in β1 and sample 25 with same directions
0.02
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sustained shifts in β1
0.01
^
sustained shifts in β1 and sample 15 with same directions
^
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shift in first profile shift in second profile
Insert figures 9-14 about here
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Figures 15 to 18 and tables 5 to 6 indicate probabilities for cases that members of the estimated ^
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covariance matrix are allowed to experience different shifts based on a factor of . For the members associated with profile correlations the shifts are considered to be both a same direction and inverse direction. The change location of samples is of a great importance. If a change takes a place on the first samples, test power index dramatically decreases. The report indicates the best performance is provided for the middle sample. Although the test power value of the last samples is allowed to decrease as well, the reduction is not as considerable as the first samples. The results shown in Table 5 indicate that test power term is allowed to increase when the changes follow inverse direction shifts. Insert figures 15-18 about here
ACCEPTED MANUSCRIPT Table 5. Standard deviation shifts for different samples shift in first ^ profile sustained shifts in
shift in second profile
for sample 5 with same directions
^
0.3 0.6 0.9 1.2 1.5
sustained shifts in for sample 5 with same directions
0.3
0.6
0.9
1.2
1.5
0.018 0.026 0.024 0.021 0.028
0.021 0.026 0.024 0.029 0.023
0.034 0.032 0.026 0.034 0.034
0.028 0.022 0.036 0.034 0.033
0.025 0.029 0.036 0.043 0.139
^
for sample 15 with same directions
^
sustained shifts in for sample 15 with same directions
0.3 0.6 0.9 1.2 1.5
0.3
0.6
0.9
0.031 0.042 0.046 0.592 0.99
0.033 0.041 0.304 0.963 1
0.042 0.304 0.945 1 1
^
^
sustained shifts in for sample 25 with same directions
1.5
0.995 1 1 1 1
and sample 25 with same directions 0.3 0.6 0.9 1.2 0.3 0.6 0.9 1.2
0.031 0.025 0.043 0.106 0.64
0.031 0.037 0.041 0.295 0.877
0.036 0.033 0.157 0.756 0.98
0.092 0.266 0.736 0.986 1
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sustained shifts in
1.2
0.587 0.969 1 1 1
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sustained shifts in
1.5
1.5
0.66 0.89 0.989 0.999 1
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Table 6. Standard deviation shifts in inverse directions for sample 15 shift in first ^ profile sustained shifts in and sample 15 with same shift in directions second profile
^
and sample 15 with inverse directions
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sustained shifts in
0.3 0.6 0.9 1.2 1.5
0.3
0.6
0.9
2.1
2.1
0.027 0.998 1 1 0.987
0.023 1 1 1
0.123 1 1 1 1
0.893 1 1 1 1
1 1 1 1 1
1
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6. Conclusion Literature addresses profile monitoring is an effective approach for analyzing the performance of a process statistically. In this approach a process is characterized by a regression equation(s). In the case that two or more response variables are addressed by a covariance matrix, the approach is referred to as multivariate profile monitoring. Multivariate profile monitoring is more complicated comparing to a single profile monitoring. The approach of monitoring technical characteristics of complicated processes is capable of supporting integration purposes corresponding to industrial information as well as approaches emphasized by Industry 4. In this paper multivariate profile monitoring for an extrusion process in phase-I of control was investigated. In this study flow rate per unit length and mass per unit area were considered as the correlated response variables. The temperature of the mold of this process was considered as the independent variable of the model. The performance of the multivariate profile model proposed in this paper was evaluated numerically comprehensively. The statistical analysis of different examples indicated that the proposed multivariate model provides a high capability of performance. In this comprehensive analysis test power term was provided.
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References [1] Xu, L. (2015), ―Enterprise integration and information architecture: a systems perspective on industrial information integration. CRC Press. [2] Chen, Y. (2016) ―Industrial information integration-a literature review 2006-2015.‖ Journal of Industrial Information Integration 2: 30-64 [3] Kang L. and Albin SL. (2000). ―On-Line Monitoring When the Process Yields a Linear Profile‖, Journal of Quality Technology, Vol. 32, أNo. 4, pp. 418-426.
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[4] Kim K, Mahmoud MA., and Woodall WH. (2003). ―On the monitoring of linear profiles‖. Journal of Quality Technology, Vol. 35, pp. 317-328. [5] Mahmoud MA., Woodall WH. (2004). "Phase I analysis of linear profiles with calibration applications". Technometrics, Vol. 46, No. 4, pp. 380-391. [6] Yeh A.B., Huwang L., Li Y.M. (2009). ―Profile monitoring for a binary response‖. IIE Transactions, Vol. 41, No. 11, pp.931–941.
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[7] Vaghefi A., Tajbakhsh S.D., and Noorossana R. (2009). ―Phase II monitoring of nonlinear profiles‖. Communications in Statistics-Theory and Methods, Vol. 38, No. 11, pp.1834-1851. [8] Kazemzadeh R.B., Noorossana R., and Amiri A. (2008). ―Phase I monitoring of polynomial profiles‖. Communications in Statistics-Theory and Methods, Vol. 37, No. 10, pp.1671-1686 [9] Amiri A., Jensen WA., Kazemzadeh RB. (2010). ―A case study on monitoring polynomial profiles in the automotive industry‖, Quality and Reliability Engineering International Vol.26, No. 3, pp. 509-520.
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[10] Colosimo BM., Pacella M., and Semeraro Q. (2008). ―Statistical process control for geometric specifications: On the monitoring of roundness profiles‖. Journal of Quality Technology, Vol.40, No. 1, pp. 1-18
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[11] Colosimo B.M., Meneses M., Semeraro Q. (2013).‖Vertical density profile monitoring using mixedeffects mode‖l 8th CIRP Conference on Intelligent Computation in Manufacturing Engineering, pp. 498 – 503.
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[12] Wang K., and Tsung F. (2005) ―Using profile monitoring techniques for a data-rich environment with huge sample size‖. Quality and reliability Engineering International. Vol. 21, No. 7, pp.677-688. [13] Zou C., Tsung F., Wang Z. (2007). ―Monitoring general linear profiles using multivariate exponentially weighted moving average schemes‖. Technometrics, Vol. 49, No. 4, pp.395-408
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[14] Mestek O., Pavlik J., Suchanek M. (1994). ―Multivariate control charts: Control charts for calibration curves‖. Fresenius Journal of Analytical Chemistry, Vol. 350, No.6, pp.344-351.
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[15] Gupta S., Montgomery DC., and Woodall WH. (2006). ―Performance evaluation of two methods for online monitoring of linear calibration profiles‖. International Journal of Production Research, Vol. 44, No. 10, pp.1927-1942. [16] Zhang J., Li Z., and Wang Z. (2009). ―Control chart based on likelihood ratio for monitoring linear profiles‖. Computational Statistics and Data Analysis, Vol. 53, No. 4, pp.1440--1448. [17] Saghaei A., Mehrjoo M., and Amiri A.( 2009). ―A CUSUM-based method for monitoring simple linear profiles‖. International Journal of Advanced Manufacturing Technology, Vol. 45, No. 11, pp.14521260. [18] Moguerza J.M., Muoz A., and Psarakis S. (2007). ―Monitoring nonlinear profiles using support vector machines‖. Lecture Notes in Computer Science, Vol. 4789, pp.574–583.
ACCEPTED MANUSCRIPT [19] Qiu P., Zou C., and Wang Z. (2010). ―Nonparametric Profile Monitoring by Mixed Effects Modeling‖. Technometrics, Vol. 52, No. 3, pp.265–277. [20] Jin J., and Shi J. (1999). ―Feature-preserving data compression of stamping tonnage information using wavelets‖. Technometrics, Vol. 41, No. 4, pp.327–339. [21] Ding Y., Zeng L., and Zhou S. (2006). ―Phase I analysis for monitoring nonlinear profiles in manufacturing processes‖. Journal of Quality Technology, Vol. 38, No. 3, pp.199–216. [22] Shang Y., Tsung F., and Zou C. (2011). ―Phase-II profile monitoring with binary data and random predictors‖. Journal of Quality Technology, Vo. 43, No. 3, pp.196–208.
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[23] Jensen WA., Birch JB., and Woodall, WH. (2008). ―Monitoring Correlation within Linear Profiles using Mixed Models‖. Journal of Quality Technology, vol. 40, No. 2, pp.167–183 [24] Atashgar K., Amir A., and Keramatee Nejad M., (2015). ―Monitoring Allan variance nonlinear profile using artificial neural network approach‖, International Journal of Quality Engineering and Technology, Vol. 5. No.2, pp.162-177. [25] Khedmati M, Niaki S.T.A. (2016). ―Phase II monitoring of general linear profiles in the presence of
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between-profile autocorrelation‖. Quality and Reliability Engineering International, Vol. 32, No. 2, pp. 443- 452. [26] Amiri A., Changliang Z., and Mohammad H. (2014). ―Monitoring Correlated Profile and Multivariate Quality Characteristics‖. Quality and Reliability Engineering International Vol. 30, No. 1, pp.133-142. [27] Noorossana R., Eyvazian M., and Vaghefi A. (2010). ―Phase II monitoring of multivariate simple linear profiles‖. Computers and Industrial Engineering, Vol. 58, No.4, pp.563–570.
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[28] Zou C., Ning X., Tsung F. (2012). ―LASSO-based multivariate linear profile monitoring‖. Annals of Operations Research, Vol. 192, No. 1, pp.3–19.
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[29] Soleimani, P., & Noorossana, R. (2014). ―Monitoring multivariate simple linear profiles in the presence of between profile autocorrelation‖. Communications in Statistics-Theory and Methods, Vol. 43, No. 3, pp.530-546.
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[30] Noorossana R., Eyvazian M., Amiri A., and Mahmoud M.A. (2010). ―Statistical monitoring of multivariate multiple linear regression profiles in Phase I with calibration application‖. Quality and Reliability Engineering International, Vol. 26, No. 3, pp.291–303. [31] Zhang, J., Ren, H., Yao, R., Zou, C., and Wang, Z. (2015). ―Phase I analysis of multivariate profiles based on regression adjustment‖. Computers & Industrial Engineering, Vol. 85, pp.132-144.
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[32] Paynabar K., Zou C. Qiu P. (2016). ―A Change Point Approach for Phase-I Analysis in Multivariate Profile Monitoring and Diagnosis‖. Technometrics, Vol. 58, No. 2, pp. 191-204. [33] Rauwendaal C. (2013). ―Polymer Extrusion‖, 5th Edition, Hanser Publishers: Munich.
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[34] Eghbali Ghahyazi M., Niaki S.T.A., Soleimani P. (2014). “On the Monitoring of Linear Profiles in Multistage Processes‖. Quality and Reliability Engineering International, Vol.30, No. 7, pp.103510475 [35] Mahmoud M.A. (2008). ―Phase I analysis of multiple linear regression profiles‖. Communications in Statistics Simulation and Computation, Vol. 37, No. 10, pp. 2106-2130. [36] Amiri A., Zand A., Soudbakhsh D. (2011). “Monitoring Simple Linear Profiles in the Leather Industry (A Case Study)‖. International Conference on Industrial Engineering and Operations Management Kuala Lumpur, Malaysia pp.891-897.
ACCEPTED MANUSCRIPT [37] Yeh A., Zerehsaz Y., (2013). ―Phase I Control of Simple Linear Profiles with Individual Observations‖. Quality and Reliability Engineering International, Vol. 29, No. 6, pp.829-840. [38] Amiri A., Koosha M., Azhdari A., Wang G., (2015). ―Phase I monitoring of generalized linear modelbased regression profiles‖. Journal of Statistical Computation and Simulation Vol.85, No.14, pp. 2839– 2859. [39] Soleimani P., Noorossana R., Amiri A. (2009). ―Simple Linear Profiles Monitoring in the Presence of Within Profile Autocorrelation‖. Computers & Industrial Engineering; Vol. 57, No. 3, pp. 1015-1021. [40] Ebadi M., Amiri A. (2012). “Evaluation of process capability in multivariate simple linear profiles‖. Scientia Iranica, Vol. 19, No. 9, pp.1960-1968
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[41] Eyvazian M., Noorossana R., Saghaei A., Amiri A. (2011). ―Phase II monitoring of multivariate multiple linear regression profiles‖. Quality and Reliability Engineering International, Vol. 27, No.3, pp. 281– 296. [42] Noorossana R., Soleimani P. (2012). ―Investigating Effect of Autocorrelation on Monitoring Multivariate Linear Profiles‖. International Journal off Industrial Engineering & Production Research, Vol. 23, No. 3, pp. 187—193
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[43] Kim, D.G. and Park, M.G. (2014), ―Horizontal Integration between cyber physical system based on Industry 4. and manufacturing execution systems through middleware building‖, Journal of Korea Multimedia Society, 17(2), pp. 1484-1493 [44] Alyahya, S., Wang, Q., Bennett, N. (2016) ―Application and integration of an RFID-enabled warehousing management system-a feasibility study.‖ Journal of Industrial Information Integration 4: 15-25.
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[45] Branger, J., Pang, Z. (2015) ―From automated home to sustainable, healthy and manufacturing home: a new story enabled by the Internet-of-Things and Industry 4.0.‖ Journal of Management Analytics 2(4): 314-332
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[46] Lee J.C., Chang T.C., and Krishnaiah P.R. (1977).‖ Approximations to the distributions of the likelihood ratio statistics for testing certain structures on the covariance matrices of real multivariate normal populations‖. Multivariate Analysis, Vol. 4, pp.105-118.
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[47] Noorossana R., Amiri A. , saghaei A. (2011). ―Statistical Analysis of Profile‖. First Edition, John Wiley: New Jersey. [48] Johnson RA, and Wichern DW.(2007). ―Applied multivariate statistical analysis‖, sixth Edition. Englewood Cliffs, NJ: Prentice hall.
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[49] Harold F., Giles J., Eldridge M. (2005) ―Extrusion: The Definitive Processing Guide and Handbook‖.
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William Andrew
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Appendix A ^
Matrix Bf :
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-0.9067 -0.9064 -0.9064 -0.9055 -0.9055 -0.9061 -0.9061 -0.9065 -0.9065 -0.9063 -0.9063 -0.9097 -0.9097 -0.9093 -0.9093 -0.9093 -0.9093 -0.911 -0.911 -0.9112 -0.9112 -0.91 -0.91 -0.9097 -0.9097 -0.9083 -0.9083 -0.9112 -0.9112 -1.8192
-0.7522 -0.7517 -0.7517 -0.7521 -0.7521 -0.7517 -0.7517 -0.7517 -0.7517 -0.7516 -0.7516 -0.7531 -0.7531 -0.7529 -0.7529 -0.7526 -0.7526 -0.7545 -0.7545 -0.7543 -0.7543 -0.755 -0.755 -0.7517 -0.7517 -0.7523 -0.7523 -0.7526 -0.7526 -1.5074
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31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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0.0164 -0.7548 -0.7548 -0.7546 -0.7546 -0.7542 -0.7542 -0.755 -0.755 -0.7554 -0.7554 -0.7561 -0.7561 -0.7553 -0.7553 -0.7555 -0.7555 -0.7551 -0.7551 -0.7555 -0.7555 -0.7543 -0.7543 -0.7528 -0.7528 -0.7543 -0.7543 -0.7541 -0.7541 -0.7522
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0.0262 -0.9131 -0.9131 -0.9126 -0.9126 -0.9146 -0.9146 -0.912 -0.912 -0.9133 -0.9133 -0.9121 -0.9121 -0.9109 -0.9109 -0.9103 -0.9103 -0.9081 -0.9081 -0.908 -0.908 -0.9072 -0.9072 -0.9065 -0.9065 -0.9072 -0.9072 -0.9074 -0.9074 -0.9067
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
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Figure1. Generic components of Extruder machine (Harold [49])
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Figure 2. Dispersion diagram of profiles correlation
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0.01 0.02 0.03 0.04
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m=15
0.05
0.03 0.02
0.01
Figure4: intercept shift for sample15
0.04
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
m=5
0.01 0.02 0.03 0.04
0.03
0.05
0.02 0.01
Figure3: intercept shift for sample 5
0.04
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
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m=25
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
0.01 0.02 0.03 0.03 0.02
0.05 0.01
Figure5: intercept shift for sample25
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0.04
0.04
m=5
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m=15
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
0.02 0.03
0.04
0.04 0.02
0.02
0.03 0.03 0.02
Figure 6: Intercept shift for inverse directions of sample 5
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Figure 7: Intercept shift for inverse directions of sample 15
m=25
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0.04
0.04 0.05 0.01
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0.05 0.01
0.03
0.01
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0.01
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
0.01 0.02 0.03
0.04
0.03
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
0.03
0.05
0.02 0.01
Figure 8: Intercept shift for inverse directions of sample 25
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m=5
0.04
0.03
0.02
0.01
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
0.01 0.02 0.03 0.04
0.03
0.04
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
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0.01 0.02 0.03 0.04 0.05
probability of signal
m=15
0.05
0.02
0.01
Figure10: Slope shift for sample 15
Figure9: Slope shift for sample 5
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m=25
0.03 0.04
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0.01 0.02
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
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0.05
0.04 0.03 0.02
0.01
Figure11: Slope shift and changes in sample 25
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m=15
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m=5
0.01 0.02 0.03 0.04 0.05 0.01
0.02
0.03
0.04
Figure13: Slope shift for inverse directions of sample15
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
0.01 0.02 0.03 0.04 0.05
0.02
0.03
0.04
0.01 Figure12: Slope shift for inverse directions of sample 5
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
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probability of signal
m=25
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
0.03
0.04
0.04 0.05 0.01
0.03 0.02
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0.01 0.02
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Figure14: Slope shift for inverse directions of sample 25
m=15
0.3 0.6 0.9
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1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1.2
1.2
0.9
1.5 0.3
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0.6
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Figure16: standard deviation shift for sample 15
1.5
m=5
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.3 0.6 1.5
0.9
1.2
1.2
0.9 0.6
1.5 0.3
Figure15: standard deviation shift for sample 5
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m=25 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.3 0.6 1.5
0.9 1.2
0.9 1.5
0.6 0.3
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1.2
Figure17: Standard deviation shifts for sample 25
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probability of signal
m=15
0.3
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0.6 0.9 1.2
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1.5
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1.5 1.2 0.9
0.6
0.3
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Figure18: standard deviation shift for inverse directions of sample 15
Monitoring Multivariate Profile Data in Plastic Parts Manufacturing Industries: An Intelligently Data Processing Karim Atashgar , Omid Asghari Zargarabadi PII: DOI: Reference:
S2452-414X(16)30094-2 10.1016/j.jii.2017.06.003 JII 41
To appear in:
Journal of Industrial Information Integration
Received date: Revised date: Accepted date:
3 November 2016 23 February 2017 18 June 2017
Please cite this article as: Karim Atashgar , Omid Asghari Zargarabadi , Monitoring Multivariate Profile Data in Plastic Parts Manufacturing Industries: An Intelligently Data Processing, Journal of Industrial Information Integration (2017), doi: 10.1016/j.jii.2017.06.003
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ACCEPTED MANUSCRIPT Monitoring Multivariate Profile Data in Plastic Parts Manufacturing Industries : An Intelligently Data Processing
Karim Atashgar1 Malek Ashtar University of Technology, Department of Industrial Engineering, Tehran 15875-1774, Iran
Omid Asghari Zargarabadi
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Email: [email protected]
Malek Ashtar University of Technology, Department of Industrial Engineering, Tehran 15875-1774, Iran
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Email: [email protected]
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Abstract: In some real cases, more complicated models such as multivariate profile, rather than simple linear profile is addressed to model the performance of the process because of the existence of correlation between the response variables. In this case, if one ignores the correlation structure of response variables by assuming separate profiles then misleading results are expected. The complicated approach, in practice, is capable of leading practitioners to analyze parameters of the performance of the process effectively. This advanced approach also helps industrial information engineers to provide the information and communication technology (ICT) effectively for real industrial cases. Extrusion is a process that is extensively used in different plastic parts manufacturing industries. Technical analysis and our statistical analysis (in this research) indicate that there are correlations between controllable technical variables of an extrusion process. In this study, two important correlated response variables, namely, flow rate per unit length and mass per unit area characteristics are considered. Monitoring jointly these two important quality factors with multivariate profile supports integration purposes of industrial information effectively. In this paper Wilks’ lambda statistic, is used for analysis of the performance of the proposed multivariate profile model in phase-I of statistical control. The performance of the proposed multivariate profile model is investigated using test power term through several different numerical cases. The proposed approach is capable of facilitating information technology (IT) activities of the studied manufacturing industry. Keywords: Multivariate profile, Statistical process control, Extrusion, Phase-I of control
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1 .Introduction In statistical process control, quality of a process or a product is monitored traditionally by a random variable or by a vector involving several correlated quality characteristics. However, really, in many practical cases, the quality of a process or a product should be described by a relationship between one or more response(s) variable(s) and one or more independent variables. This advanced regression approach is referred to as the profile monitoring. The profile approach provides more realistic capability of monitoring product/ process characteristics comparing to conventional methods. The new approach can be attended in the second level of the discipline structure of industrial information integration engineering (IIIE) introduced by Xu [1]. 1
Corresponding author. Tel/Fax: +982122970309 E-mail address: [email protected] (K. Atashgar)
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Chen [2] also emphasized on new approaches for manufacturing industries by a comprehensive literature review of industrial information integration. In recent years different researchers have reported several studies approaching profile. Several authors such as Kang and Albin [3], Kim et al. [4], Mahmoud and Woodall [5], Yeh et al. [6], Vaghefi et al. [7], Kazamzadeh et al. [8],Amiri et al. [9] ,Colosimo et al. [10,11], Wang and Tsung [12], Zou et al. [13], Mestek et al. [14], Gupta et al. [15], Zhang et al. [16] , Saghaei et al. [17], Moguerza et al. [18], Qui et al. [19], Jin and Shi [20], Ding et al. [21], Shang et al. [22], Jensen et al. [23], Atashgar et al. [24], Khedmati and Niaki [25] have contributed to linear and nonlinear univariate profile monitoring. Literature addresses that Amiri et al. [26], Noorossana et al. [27], Zou et al. [28] and Soleimani and Noorossan [29], focused on monitoring a multivariate profile of phase-II. In this arena Noorossana et al. [30], Zhang et al. [31], and Paynabar et al. [32] have contributed to phase-I of multivariate profile monitoring. Noorossana et al. [30] considered two correlated regression equations in order to generate simulated data. In this consideration they generated simulated data based on assumed marginal normal distribution of each response variable. They assumed that there is an assumed covariance matrix and the distribution of the correlated response variables follows a bivariate normal because of simulated data of each response variable follows a marginal normal distribution. The approach used by Noorossana et al. [30] may leads to misleading signals. The bivariate normal test of the joint response variables of the example has not been also reported by Noorossana et al. [30]. The Noorossana et al. [30] report provides an evaluation analysis without regarding inverse shift directions. In Table 1 the literature is summarized. As shown in Table 1 univariate type of profile is focused on by most authors. Extruder process is used extensively in different plastic parts manufacturing industries. An effective method to monitor this process considerably helps practitioners to control technical factors of the process. Extruder is a machine which takes raw materials mostly consisting of rubber or plastic polymers and then produces products in different shapes. In this process two important quality specifications i.e. flow rate per unit length and mass per unit area should be controlled by quality engineers. Technical analysis indicates that flow rate per unit length and mass per unit area are affected by mold temperatures (see Rauwendaal [33].The technical report of Rauwendaal [33] indicates that the two specifications are also correlated. Table 1: Profile resources of literature
Phase-I
CE Phase-II
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No. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 16 17 18 19 20 21
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Phase type
Mahmoud &Woodall [5] Ding Y et al [21] Gupta et al [15] Moguerza et al [18] Eghbali et al [34] Kazemzadeh et al [8] Mahmoud [35] Colosimo et al. [10] Noorossana et al [30] Amiri et al [9] Amiri et al. [36] Yeh & Zerehsaz [37] Amiri et al [38] Zhang et al. [31] Atashgar et al [24] Paynabar et al. [32] Jin & Shi [20] Kang & Albin [3] Kim et al [4] Saghaei et al. [17] Soleimani et al. [39] Vaghefi et al. [7]
Profile type Univariate Multivariate ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
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Jensen et al. [23] Colosimo et al [11]
● ● ● ● ●
Ebadi & Amiri [40] Eyvazian et al. [41] Noorossana & Soleimani [42] Soleimani & Noorossana [29] Khedmati & Niaki[ [25]
● ● ●
Amiri et al. [26]
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This study focuses on monitoring flow rate per unit length and mass per unit technical characteristics using profile approach. This research analyzes the performance of the proposed multivariate profile for phase-I. The statistical analysis of this study confirms the existence of correlation between flow rate per unit length and mass per unit claimed in the technical analysis of Rauwendaal [33]. In this study the mold temperature of the extrusion process is allowed to play as an independent variable for the multivariate profile. This approach provides the required capability of a smart factory for plastic parts manufacturing industries basically. Kim and Parek [43] analyzed Industry 4.for small batch production. The investigation conveys the idea of the integration production management systems (i.e. technical characteristics monitoring and etc.) with developing new technologies. The profile approach leads to accuracy of inventory data as emphasized by Alyahya et al. [44]. Branger and Pang [45] discussed about the domain of the vision of future homes including automated home, sustainable home, healthy home, and manufacturing home. The profile monitoring can be used in all sections of the future homes.
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The rest of this paper is organized as follows: Section 2 introduces the studied extrusion process. Multivariate profile monitoring is discussed in section 3. Our proposed model to control the performance of an extruder and statistical analyzes are explicated in section 4. The evaluation report and our concluding remarks are provided in sections 5 and 6, respectively.
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2. Briefing the extruder process An extruder machine is used to produce different plastic/ rubber parts. In this process a compound is passed through a mold under a high pressure, so that the material is allowed to form in different cross sections. In the studied extruder process, components are produced continuously and the input is polymer material (PE). In this process, the output is produced in the shape of thin sheet continuously. In this process uniform mass distribution over the sheet and flow rate per unit length of the sheet are two crucial technical characteristics. Figure 1 addresses the main components of the extruder machine. Insert Figure 1 about here
In this process mold temperature is also a crucial specification. When temperature of a mold decreases or increases, viscosity of material is allowed to change. Rauwendaal [33] explained that flow rate per unit length and mass per unit area characteristics are two correlated variables. As addressed by Rauwendaal [33] dynamics of the output flow of an extruder is influenced directly by the mold temperature. The flow dynamics affects both flow rate per unit length and mass per unit area factors. In this paper the correlation of these two variables is also addressed statistically. In this paper to control the performance of an extruder machine, a multivariate profile model is proposed. In addition the performance analysis of this process is provided for phase-I of statistical quality control. 3. Multivariate profiles
ACCEPTED MANUSCRIPT Generally, linear profiles can be classified into two branches; 1) single response variable, and 2) multi correlated response variables (multivariate profile). In a multivariate profile there are more than one response variable. In this case each response is defined as a function of an independent variable(s), while the response variables mutually affect each other. Assume that there are m sample sets for an independent variable x, each of which includes n observations, and there are p response variables for each independent variable. In this case there are n observations for the kth sample as follows. (x i , yi 1k , yi 2 k ,...., y ipk )
(1)
Yk XBk Ek y 12 k y 22 k y n 2k
where
y 1 pk 1 x 1 1 x y 2 pk 01k 2 11k y npk 1 x n
Yk (y1k , y 2 k ,..., y nk )T is
02 k 12 k
0 pk 1 pk
11k 21k n 1k
12 k 22 k
n 2k
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y 11k y 21k y n 1k
(2)
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where x i indicates the ith sample value from the jth sampling for p response variables of y i 1k , y i 2k ,..., y ipk . For each sample, the model addresses a relationship between each response variable and an independent variable as follows. 1 pk 2 pk npk
a n p response variables matrix for the k sample. Each row of th
(2)
Yk
corresponds to one value of independent variable ( x i ) and includes p values for response variables. Let
n 2 dimension, where x includes the values of the denotes a n p error matrix for the kth sample. The rows of
X [1 x] be given as an independent matrix with a
independent variable. Matrix
Ek (ε1k , ε 2 k ,...ε nk )
T
M
Ek follow independently an identical multivariate normal distribution with 1 p zero mean vector and p p covariance matrix Σ . In another word, elements of each row of Ek are dependent; however, each row is
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addressed independently. In Equation 2, coefficient matrix
Bk (β0 k , β1k )T
denotes a 2 p dimension. This
ˆ )T (Y XB ˆ ) as the following matrix is estimated from least squares method which minimizes (Yk XB k k k
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equation.
y .1k ˆ1 jk x Sxy (1) Sxx
ˆ0 pk ˆ1 pk
y . pk ˆ1 jk x Sxy ( p ) Sxx
(3)
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where
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ˆ ˆ (XT X)-1 XT Y (βˆ , βˆ ) T 01k B k k 0k 1k ˆ 11k
n n 1 n 1 n x x i ، S xy ( j ) (x i x ) y ijk ، y . jk y ijk ، S xx (x i x )2 n i 1 n i 1 i 1 i 1
(4)
3.1. Wilks’ lambda statistic approach In this research Wilk’s statistic is used to analyze the proposed model. The statistic is developed based on F-statistic proposed by Mahmoud and Woodall [5]. Mahmoud and Woodall [5] proposed the use of indicator or artificial variables for a multiple regression. The method is developed for a multivariate regression. Consider m samples each of which is included of n observations as follows: k =1,2,…,m; i = 1,2,…,n. (x ik , y i 1k , y i 2 k ,..., y ipk ) (5)
ACCEPTED MANUSCRIPT The aim is to examine the equality of regressions of different samples. For this purpose, all m samples are pooled and a sample set including mn members is constructed. In this stage it is possible to define m-1 indicator variables as follows. {
.
(6)
In this case it is allowed to define the following test statistic of null hypothesis H 0 . ˆ X Y YY B f f . ˆ YY Br Xr Y where
Y (y1 , y 2 ,..., y mn ) T
(7)
is a mn p matrix of response variables for the pooled variables. In Equation 7,
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Xf denotes a mn 2m matrix of independent variables of the full model, and Bˆ f indicates a 2m p matrix including estimated coefficients of the full regression model. However, in this equation Xr with mn 2
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dimension and Bˆ r with 2 p dimension address the reduced model. Under a null hypothesis, this statistic follows Wilks’ lambda distribution with p, 2(m-1) and m (n-2) degrees of freedom. The null hypothesis is rejected when the test statistic is less than , p ,2( m 1),m ( n 2) . In this research, for the equality test of a covariance matrix, the method proposed by Lee et al. [46] is utilized. For this purpose, unbiased estimation of covariance matrix for the kth sample is denoted as follows.
Yk Yk Bˆ k X Yk . (8) n 2 where Yk is a n p matrix of response variables, X [1 x] is a n 2 matrix of independent variable, and Bˆ k is Sk
M
the estimation of least squares. In order to generate the test statistic, M is calculated as the below equation 9. ( n 2)/2 ( n 2)/2 ( n 2)/2 S1 S2 Sm ( S 1 S 2 S m )( n 2)/2 M m( n 2)/2 m( n 2)/2 . (9) S S S pl is
m
S pl
k 1 m
the pooled value of the covariance matrix which is determined as follows. (n 2)S k
(n 2) k 1
m k 1
Sk
m
(10)
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where
pl
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pl
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When M value is allowed to be close to 1, it is addressed that the equality hypothesis is not rejected, whereas in the case that the value is close to zero value, it is allowed to reject the hypothesis. The reader for exact critical value of -2lnM statistic is referred to Lee et al. [46]. Noorossana et al. [47] addressed that the developed Fstatistic method called Wilk’s statistic is superior compared to other method proposed in literature of multivariate cases. In this paper, this method is used to monitor the proposed multivariate profile of the extruder process. 4. Phase I extruder profiles monitoring using Wilks’ lambda statistic method As mentioned before flow rate per unit length and mass per unit area are considered as two important response variables in the process of extruder, while the mold temperature characteristic is an explanatory variable in this process. In this research a digital displayer and a digital scale are employed to record data of the flow rate per unit length and the mass per unit area respectively. Response variable values are provided for 11 different levels (220 to 240 with distances 2). Since the temperature factor differs point by point of a mold; hence, in this study the averaged measure is utilized. Considering aforementioned information and 3 repetitive sampling in each level of independent variable, in this case n (the size of a sample) is addressed by Noorossana et al. [47]. The sampling is done 30 times i.e. m equals to 30, and overall, there are 30 samples each of which
ACCEPTED MANUSCRIPT includes 33 observations leading to 990 random experiments. To calculate Wilk’s lambda statistic , Bˆ f and
Bˆ r are obtained from Equation 11. Bˆ r (XT X)-1 XT Yr , Bˆ (XT X)-1 XT Y f
(11)
f
1.8187 1.5074 ˆ Calculations lead to Bˆ r , however Bf is a 2 60 matrix which is provided in the appendix. 0.0262 0.0164 In this study Wilk’s statistics equals to 0.86675 . Since , p ,2( m 1),m ( n 2) 0.025320,2,58,930 0.85736 0.86675 is addressed, the null hypothesis is not rejected. It means that the process is in-control in phase-I.
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2
Table2. Covariance matrix of each profile
S2 S3 S4 S5 S6
0.8528
0.0069
0.0037
1.5802
0.8522
0.0069
0.0037
1.5726
0.8493
0.0068
0.0037
1.4269
0.7690
0.0062
0.0033
1.4091
0.7596
0.0061
0.0033
1.4490
0.7802
0.0063
0.0034
S7 S8 S9 S10 S11 S12
1.5855
0.8539
0.0069
0.0037
1.6030
0.8629
0.0070
0.0037
1.6077
0.8648
0.0070
0.0038
1.6660
0.8959
0.0072
0.0039
1.6227
0.8731
0.0070
0.0038
1.6203
0.8727
0.0070
0.0038
S13 S14 S15 S16 S17 S18
1.6117
0.8672
-0.0070
0.0038
1.6234
0.8737
-0.0071
0.0038
1.6172
0.8716
-0.0070
0.0038
1.7075
0.9205
-0.0074
0.0040
S19 S20 S21 S22
1.6285
0.8780
-0.0071
0.0038
1.6903
0.9114
-0.0073
0.0040
1.6038
0.8650
-0.0070
0.0038
1.6318
0.8800
-0.0071
0.0038
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S1
1.5811
1.6547
0.8914
-0.0072
0.0039
1.7291
0.9321
-0.0075
0.0040
S23 S24
1.5939
0.8606
-0.0069
0.0037
1.4195
0.7650
-0.0062
0.0033
S25 S26 S27 S28 S29 S30
1.4514
0.7824
0.0063
-0.0034
1.4263
0.7679
0.0062
-0.0033
1.6219
0.8757
0.0070
-0.0038
1.6037
0.8649
0.0070
-0.0038
1.5939
0.8606
0.0069
-0.0037
1.4387
0.7754
0.0062
-0.0034
M
As discussed before to test the covariance matrix equality, the method proposed by Lee et al. [46] is used. The results are listed in Table1 for 30 samples of this study. Each S ( ) shown in Table 2 addresses the variance-covariance matrix of each m sample. The calculation leads to the following matrix for S pl .
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1.57905 0.846245 S pl 0.00686 0.00370 In this step using Equation 9, M is derived to be M 0.985185185 . Since M statistic is very close to 1, the covariance equality hypothesis is confirmed.
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5. Proposed Model evaluation In this evaluation the covariance matrix of the multivariate regression is calculated using Equation 12 (where N and r respectively denote total number of samples and total number of estimated parameters). This matrix allows us to produce simulated data for the studied process. In this paper the following cases are considered for analyzing the proposed profiles: 1) Synchronous and asynchronous shift cases. 2) Shifts with the same and inverse directions corresponding to parameters of the multivariate profile. To address the normality of the correlated response variables Johnson and Wichern [48] method is used. To evaluate the performance of the proposed multivariate profile each combination of shift size is simulated for 10000 iterations. In this evaluation, equations 12 and 13 obtained from phase-I of the extruder process are utilized. In these equations n=33, m=30 and ( i 1k , i 2k . i 3k , i 4k ) follow multivariate normal distribution with zero mean vector and covariance of . The estimation of is estimated from Equation 12.
ACCEPTED MANUSCRIPT ^
^
^
^
ε ε (Y-Z )' (Y-Z ) 0.5599 0.1361 3 10 0.1361 0.199 3 N-r-1 N-r-1 ^
(12)
Equation13 addresses the fitted multivariate profile model based on data for phase-I of the extruder process.
Y i 1k 1.8187 0.0262 x i i 1k
(13)
i 1, 2,, n , k 1, 2, , m
Y i 2k 1.5074 0.0164 x i i 2k
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Figure 2 shows the correlation of profile versus profile using simulated data obtained from 10000 iterations. As shown in Figure 2 the correlation between the profiles is negative. It should be stated that the variance of the first profile is larger compared to the second one. In this paper to evaluate the proposed model the following shifts are examined: 1) Shift in β 0 with different amount values, for various samples and in different directions (i.e. an intercept parameter decreases while another one is allowed to increase). 2) Shift in β1 with different amount values, for various samples and in different directions (i.e. a slope increases while another one decreases).
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^
3) Different shift values in for different samples and in different directions (i.e. one of the variances increases while another one is allowed to decrease). In this analysis type-I error probability for Wilks’ lambda statistic and M statistic of Lee et al. [46] is assumed to be α 2 1 1 α 0.02532 so that the total type-I error becomes 0.05. These shifts are ^
simultaneously induced into profiles 1 and 2. The intercept shift is in (1/ n ) (X 2 / s xx ) unit, the shift of
M
^
^
slope is in / s xx unit, and the shift values of covariance are in a factor of .
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Insert Figure 2 about here
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5.1. Normality test of response variables If several variables follow a joint distribution such as a multivariable normal distribution, the marginal distribution of each of the variables follows single normal distribution; nevertheless, the inverse is not addressed necessarily. In other words, the joint distribution of the variables does not necessarily follow a normal multivariable distribution in the case that each distribution of variables is addressed by a normal distribution (for more details, see Johnson and Wichern [48]). Hence if each row of Y follows for example a bivariate 2 normal distribution, it is expected that χ p, 0.0201 with error value 0.01 , and the number of dependent variables p=2. In this case it is expected that 99% of a desired bivariate distribution points are located inside the ellipse represented by Equation 14. (Y-Y)' S-1 (Y-Y) χ 22,0.01 (14) Now for the case of this research the following equation is obtained: /
y 1 4.20 0.0279 0.017 y 1 4.20 2 χ 2,0.01 0.0201 y 2 2.26 0.017 0.011 y 2 2.26 To check normality of the studied multivariate profile, Equation 14 is coded. The results revealed that normality hypothesis is acceptable. 5.2. Test power for intercept, slop, and covariance shifts
ACCEPTED MANUSCRIPT Figures 3 to 8 and Table 2 provide out-of-control signal probability for different shifts from 10000 iterations. The results indicate a high capability for the proposed model when the intercept parameter is affected by different shift sizes. As shown in Table 2 the best performance is addressed when a change occurs for the middle sample (that is to say, the best performance is obtained when shifts occur in sample 15). When changes get far from the middle sample, the performance addresses degradation, so that there is symmetry between performance degradation in the first and the last samples. It should be stated that test power index is allowed to decrease when a same direction of shifts are experienced by the process. It is interesting that even in this case, the best test power value is addressed by the middle sample.
shift in first profile
shift in first profile
^
sustained shifts in β 0 for sample 5
shift in second profile
shift in second profile
with same directions
sustained shifts in β 0 for sample 5 with same directions
0.01 0.02 0.03 0.03 0.04 0.03 0.05
0.01 0.063 0.126 0.312 0.615 0.885
0.02 0.122 0.273 0.53 0.838 0.959
0.03 0.299 0.527 0.786 0.935 0.999
0.04 0.616 0.799 0.947 0.995 1
^
sustained shifts in β 0 for sample 15 with same directions
sustained shifts in β 0 for sample 15 with same directions
0.01 0.02 0.03 0.04 0.05
0.01 0.076 0.282 0.688 0.948 0.997
0.02 0.304 0.607 0.903 0.996 1
0.03 0.69 0.906 0.996 1 1
^
0.04 0.968 0.998 1 1 1
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for sample 25 with same directions
0.01 0.058 0.115 0.296 0.653 0.887
0.02 0.139 0.279 0.524 0.83 0.953
0.03 0.303 0.539 0.75 0.947 0.996
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sustained shifts in β 0
0.01 0.02 0.03 0.04 0.05
0.04 0.611 0.809 0.939 0.994 0.998
for sample 5 with inverse directions
0.01 0.02 0.03 0.03 0.04 0.03 0.05
0.01 0.04 3 0.07 3 0.15 1 0.35 4 0.66 2
0.02 0.07 4 0.09 6 0.16 2 0.37 1 0.61 4
0.03 0.15 2 0.14 8 0.24 6 0.42 4 0.68 3
0.04 0.35 8 0.34 2 0.44 3 0.58 0.75 5
0.05 0.66 8 0.63 9 0.67 5 0.75 6 0.86
sustained shifts in β 0 for sample 15 with opposite directions
0.05 1 1 1 1 1
sustained shifts in β 0 and sample 25 with same directions ^
^
sustained shifts in β 0
^
M
^
0.05 0.879 0.965 0.994 0.998 1
^
sustained shifts in β 0 for sample 5 with inverse directions
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^
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Table3. Intercept shift values in different locations and the same and inverse directions
0.05 0.885 0.964 0.994 0.999 1
^
sustained shifts in β 0 for sample 15 with inverse directions
0.01 0.02 0.03 0.04 0.05
^
0.01 0.04 1 0.11 3 0.35 5 0.73 9 0.97 5
0.02 0.1 0.19 0.37 7 0.74 7 0.95 7
0.03 0.36 0.37 6 0.55 8 0.83 1 0.97 8
0.04 0.78 4 0.76 2 0.83 9 0.94 6 0.99
0.05 0.97 0.96 3 0.97 3 0.98 8 1
sustained shifts in β 0 and sample 25 with inverse directions ^
sustained shifts in β 0 for sample 25 with inverse directions
0.01 0.02 0.03 0.04 0.05
0.01 0.04 6 0.05 3 0.14 8 0.35 9 0.71
0.02 0.07 2 0.08 8 0.17 5 0.33 0.62
0.03 0.15 8 0.16 3 0.25 9 0.42 9 0.66
0.04 0.37 7 0.32 4 0.42 5 0.56 0.76
0.05 0.66 8 0.65 0.70 1 0.75 9 0.88
Insert figures 3-8 about here ^
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The following figures and Table 3 provide out-of-control signals when β1 parameter experiences ^
^
different sustain shifts i.e. β1 / S xx . As shown in Table 4 the best performance is addressed in the middle sample i.e. the 15th sample. When changes get far from the middle sample, the performance is allowed to degrade. In this case there is symmetry between performance degradation. It should be stated that the test power term decreases when slope shifts are not addressed by a same direction.
ACCEPTED MANUSCRIPT Table4. Slope shift values in different locations and in the same and inverse directions shift in first ^ profile ^ sustained shifts in β1 for sample 5 shift in sustained shifts in β1 for sample 5 second profile with same directions with inverse directions
0.01
0.02
0.03
0.04
0.05
0.01
0.05
0.11
0.32
0.62
0.87
sustained shifts in β1
0.02
0.13
0.25
0.52
0.81
0.96
for sample 5 with same directions
0.03
0.30
0.52
0.76
0.93
0.99
0.04
0.60
0.81
0.93
0.99
0.99
0.05
0.88
0.95
0.99
1
1
^
^
for sample 15 with same directions
0.01
0.02
0.03
0.04
0.05
0.11
0.26
0.68
0.95
0.99
0.02
0.29
0.60
0.90
0.99
0.99
0.03
0.67
0.91
0.98
1
1
0.04
0.93
0.99
1
1
1
0.05
0.99
1
1
1
1
^
0.01
0.02
0.03
0.04
0.01
0.05
0.11
0.29
0.6
sustained shifts in β1
0.02
0.11
0.25
0.50
0.79
for sample 25 with same directions
0.03
0.30
0.52
0.75
0.95
0.04
0.62
0.80
0.94
0.05
0.87
0.95
0.99
0.05
0.01
0.03
0.06
0.13
0.35
0.65
sustained shifts in β1
0.02
0.07
0.08
0.15
0.31
0.62
for sample 5 with inverse directions
0.03
0.12
0.15
0.23
0.43
0.66
0.04
0.34
0.35
0.39
0.56
0.75
0.05
0.67
0.63
0.64
0.75
0.86
^
^
sustained shifts in β1
for sample 15 with inverse directions
0.01
0.02
0.03
0.04
0.05
0.01
0.05
0.11
0.37
0.75
0.97
0.02
0.09
0.19
0.38
0.73
0.97
0.03
0.32
0.39
0.55
0.83
0.97
0.04
0.74
0.73
0.82
0.93
0.99
0.05
0.96
0.96
0.97
0.98
0.99
^
0.05
0.01
0.02
0.03
0.04
0.05
0.87
0.01
0.03
0.06
0.13
0.37
0.66
^
0.95
sustained shifts in β1
0.02
0.06
0.07
0.18
0.36
0.62
0.99
for sample 25 with inverse directions
0.03
0.14
0.14
0.21
0.38
0.66
0.04
0.35
0.34
0.42
0.50
0.73
0.05
0.66
0.62
0.65
0.75
0.86
0.99
0.99
0.99
1
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^
0.04
sustained shifts in β1 and sample 25 with inverse directions
M
0.03
sustained shifts in β1 and sample 15 with inverse directions
0.01
sustained shifts in β1 and sample 25 with same directions
0.02
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sustained shifts in β1
0.01
^
sustained shifts in β1 and sample 15 with same directions
^
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shift in first profile shift in second profile
Insert figures 9-14 about here
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Figures 15 to 18 and tables 5 to 6 indicate probabilities for cases that members of the estimated ^
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covariance matrix are allowed to experience different shifts based on a factor of . For the members associated with profile correlations the shifts are considered to be both a same direction and inverse direction. The change location of samples is of a great importance. If a change takes a place on the first samples, test power index dramatically decreases. The report indicates the best performance is provided for the middle sample. Although the test power value of the last samples is allowed to decrease as well, the reduction is not as considerable as the first samples. The results shown in Table 5 indicate that test power term is allowed to increase when the changes follow inverse direction shifts. Insert figures 15-18 about here
ACCEPTED MANUSCRIPT Table 5. Standard deviation shifts for different samples shift in first ^ profile sustained shifts in
shift in second profile
for sample 5 with same directions
^
0.3 0.6 0.9 1.2 1.5
sustained shifts in for sample 5 with same directions
0.3
0.6
0.9
1.2
1.5
0.018 0.026 0.024 0.021 0.028
0.021 0.026 0.024 0.029 0.023
0.034 0.032 0.026 0.034 0.034
0.028 0.022 0.036 0.034 0.033
0.025 0.029 0.036 0.043 0.139
^
for sample 15 with same directions
^
sustained shifts in for sample 15 with same directions
0.3 0.6 0.9 1.2 1.5
0.3
0.6
0.9
0.031 0.042 0.046 0.592 0.99
0.033 0.041 0.304 0.963 1
0.042 0.304 0.945 1 1
^
^
sustained shifts in for sample 25 with same directions
1.5
0.995 1 1 1 1
and sample 25 with same directions 0.3 0.6 0.9 1.2 0.3 0.6 0.9 1.2
0.031 0.025 0.043 0.106 0.64
0.031 0.037 0.041 0.295 0.877
0.036 0.033 0.157 0.756 0.98
0.092 0.266 0.736 0.986 1
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sustained shifts in
1.2
0.587 0.969 1 1 1
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sustained shifts in
1.5
1.5
0.66 0.89 0.989 0.999 1
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Table 6. Standard deviation shifts in inverse directions for sample 15 shift in first ^ profile sustained shifts in and sample 15 with same shift in directions second profile
^
and sample 15 with inverse directions
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sustained shifts in
0.3 0.6 0.9 1.2 1.5
0.3
0.6
0.9
2.1
2.1
0.027 0.998 1 1 0.987
0.023 1 1 1
0.123 1 1 1 1
0.893 1 1 1 1
1 1 1 1 1
1
AC
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6. Conclusion Literature addresses profile monitoring is an effective approach for analyzing the performance of a process statistically. In this approach a process is characterized by a regression equation(s). In the case that two or more response variables are addressed by a covariance matrix, the approach is referred to as multivariate profile monitoring. Multivariate profile monitoring is more complicated comparing to a single profile monitoring. The approach of monitoring technical characteristics of complicated processes is capable of supporting integration purposes corresponding to industrial information as well as approaches emphasized by Industry 4. In this paper multivariate profile monitoring for an extrusion process in phase-I of control was investigated. In this study flow rate per unit length and mass per unit area were considered as the correlated response variables. The temperature of the mold of this process was considered as the independent variable of the model. The performance of the multivariate profile model proposed in this paper was evaluated numerically comprehensively. The statistical analysis of different examples indicated that the proposed multivariate model provides a high capability of performance. In this comprehensive analysis test power term was provided.
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References [1] Xu, L. (2015), ―Enterprise integration and information architecture: a systems perspective on industrial information integration. CRC Press. [2] Chen, Y. (2016) ―Industrial information integration-a literature review 2006-2015.‖ Journal of Industrial Information Integration 2: 30-64 [3] Kang L. and Albin SL. (2000). ―On-Line Monitoring When the Process Yields a Linear Profile‖, Journal of Quality Technology, Vol. 32, أNo. 4, pp. 418-426.
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[4] Kim K, Mahmoud MA., and Woodall WH. (2003). ―On the monitoring of linear profiles‖. Journal of Quality Technology, Vol. 35, pp. 317-328. [5] Mahmoud MA., Woodall WH. (2004). "Phase I analysis of linear profiles with calibration applications". Technometrics, Vol. 46, No. 4, pp. 380-391. [6] Yeh A.B., Huwang L., Li Y.M. (2009). ―Profile monitoring for a binary response‖. IIE Transactions, Vol. 41, No. 11, pp.931–941.
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[7] Vaghefi A., Tajbakhsh S.D., and Noorossana R. (2009). ―Phase II monitoring of nonlinear profiles‖. Communications in Statistics-Theory and Methods, Vol. 38, No. 11, pp.1834-1851. [8] Kazemzadeh R.B., Noorossana R., and Amiri A. (2008). ―Phase I monitoring of polynomial profiles‖. Communications in Statistics-Theory and Methods, Vol. 37, No. 10, pp.1671-1686 [9] Amiri A., Jensen WA., Kazemzadeh RB. (2010). ―A case study on monitoring polynomial profiles in the automotive industry‖, Quality and Reliability Engineering International Vol.26, No. 3, pp. 509-520.
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[10] Colosimo BM., Pacella M., and Semeraro Q. (2008). ―Statistical process control for geometric specifications: On the monitoring of roundness profiles‖. Journal of Quality Technology, Vol.40, No. 1, pp. 1-18
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[11] Colosimo B.M., Meneses M., Semeraro Q. (2013).‖Vertical density profile monitoring using mixedeffects mode‖l 8th CIRP Conference on Intelligent Computation in Manufacturing Engineering, pp. 498 – 503.
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[12] Wang K., and Tsung F. (2005) ―Using profile monitoring techniques for a data-rich environment with huge sample size‖. Quality and reliability Engineering International. Vol. 21, No. 7, pp.677-688. [13] Zou C., Tsung F., Wang Z. (2007). ―Monitoring general linear profiles using multivariate exponentially weighted moving average schemes‖. Technometrics, Vol. 49, No. 4, pp.395-408
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[14] Mestek O., Pavlik J., Suchanek M. (1994). ―Multivariate control charts: Control charts for calibration curves‖. Fresenius Journal of Analytical Chemistry, Vol. 350, No.6, pp.344-351.
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[15] Gupta S., Montgomery DC., and Woodall WH. (2006). ―Performance evaluation of two methods for online monitoring of linear calibration profiles‖. International Journal of Production Research, Vol. 44, No. 10, pp.1927-1942. [16] Zhang J., Li Z., and Wang Z. (2009). ―Control chart based on likelihood ratio for monitoring linear profiles‖. Computational Statistics and Data Analysis, Vol. 53, No. 4, pp.1440--1448. [17] Saghaei A., Mehrjoo M., and Amiri A.( 2009). ―A CUSUM-based method for monitoring simple linear profiles‖. International Journal of Advanced Manufacturing Technology, Vol. 45, No. 11, pp.14521260. [18] Moguerza J.M., Muoz A., and Psarakis S. (2007). ―Monitoring nonlinear profiles using support vector machines‖. Lecture Notes in Computer Science, Vol. 4789, pp.574–583.
ACCEPTED MANUSCRIPT [19] Qiu P., Zou C., and Wang Z. (2010). ―Nonparametric Profile Monitoring by Mixed Effects Modeling‖. Technometrics, Vol. 52, No. 3, pp.265–277. [20] Jin J., and Shi J. (1999). ―Feature-preserving data compression of stamping tonnage information using wavelets‖. Technometrics, Vol. 41, No. 4, pp.327–339. [21] Ding Y., Zeng L., and Zhou S. (2006). ―Phase I analysis for monitoring nonlinear profiles in manufacturing processes‖. Journal of Quality Technology, Vol. 38, No. 3, pp.199–216. [22] Shang Y., Tsung F., and Zou C. (2011). ―Phase-II profile monitoring with binary data and random predictors‖. Journal of Quality Technology, Vo. 43, No. 3, pp.196–208.
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[23] Jensen WA., Birch JB., and Woodall, WH. (2008). ―Monitoring Correlation within Linear Profiles using Mixed Models‖. Journal of Quality Technology, vol. 40, No. 2, pp.167–183 [24] Atashgar K., Amir A., and Keramatee Nejad M., (2015). ―Monitoring Allan variance nonlinear profile using artificial neural network approach‖, International Journal of Quality Engineering and Technology, Vol. 5. No.2, pp.162-177. [25] Khedmati M, Niaki S.T.A. (2016). ―Phase II monitoring of general linear profiles in the presence of
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between-profile autocorrelation‖. Quality and Reliability Engineering International, Vol. 32, No. 2, pp. 443- 452. [26] Amiri A., Changliang Z., and Mohammad H. (2014). ―Monitoring Correlated Profile and Multivariate Quality Characteristics‖. Quality and Reliability Engineering International Vol. 30, No. 1, pp.133-142. [27] Noorossana R., Eyvazian M., and Vaghefi A. (2010). ―Phase II monitoring of multivariate simple linear profiles‖. Computers and Industrial Engineering, Vol. 58, No.4, pp.563–570.
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[28] Zou C., Ning X., Tsung F. (2012). ―LASSO-based multivariate linear profile monitoring‖. Annals of Operations Research, Vol. 192, No. 1, pp.3–19.
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[29] Soleimani, P., & Noorossana, R. (2014). ―Monitoring multivariate simple linear profiles in the presence of between profile autocorrelation‖. Communications in Statistics-Theory and Methods, Vol. 43, No. 3, pp.530-546.
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[30] Noorossana R., Eyvazian M., Amiri A., and Mahmoud M.A. (2010). ―Statistical monitoring of multivariate multiple linear regression profiles in Phase I with calibration application‖. Quality and Reliability Engineering International, Vol. 26, No. 3, pp.291–303. [31] Zhang, J., Ren, H., Yao, R., Zou, C., and Wang, Z. (2015). ―Phase I analysis of multivariate profiles based on regression adjustment‖. Computers & Industrial Engineering, Vol. 85, pp.132-144.
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[32] Paynabar K., Zou C. Qiu P. (2016). ―A Change Point Approach for Phase-I Analysis in Multivariate Profile Monitoring and Diagnosis‖. Technometrics, Vol. 58, No. 2, pp. 191-204. [33] Rauwendaal C. (2013). ―Polymer Extrusion‖, 5th Edition, Hanser Publishers: Munich.
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[34] Eghbali Ghahyazi M., Niaki S.T.A., Soleimani P. (2014). “On the Monitoring of Linear Profiles in Multistage Processes‖. Quality and Reliability Engineering International, Vol.30, No. 7, pp.103510475 [35] Mahmoud M.A. (2008). ―Phase I analysis of multiple linear regression profiles‖. Communications in Statistics Simulation and Computation, Vol. 37, No. 10, pp. 2106-2130. [36] Amiri A., Zand A., Soudbakhsh D. (2011). “Monitoring Simple Linear Profiles in the Leather Industry (A Case Study)‖. International Conference on Industrial Engineering and Operations Management Kuala Lumpur, Malaysia pp.891-897.
ACCEPTED MANUSCRIPT [37] Yeh A., Zerehsaz Y., (2013). ―Phase I Control of Simple Linear Profiles with Individual Observations‖. Quality and Reliability Engineering International, Vol. 29, No. 6, pp.829-840. [38] Amiri A., Koosha M., Azhdari A., Wang G., (2015). ―Phase I monitoring of generalized linear modelbased regression profiles‖. Journal of Statistical Computation and Simulation Vol.85, No.14, pp. 2839– 2859. [39] Soleimani P., Noorossana R., Amiri A. (2009). ―Simple Linear Profiles Monitoring in the Presence of Within Profile Autocorrelation‖. Computers & Industrial Engineering; Vol. 57, No. 3, pp. 1015-1021. [40] Ebadi M., Amiri A. (2012). “Evaluation of process capability in multivariate simple linear profiles‖. Scientia Iranica, Vol. 19, No. 9, pp.1960-1968
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[41] Eyvazian M., Noorossana R., Saghaei A., Amiri A. (2011). ―Phase II monitoring of multivariate multiple linear regression profiles‖. Quality and Reliability Engineering International, Vol. 27, No.3, pp. 281– 296. [42] Noorossana R., Soleimani P. (2012). ―Investigating Effect of Autocorrelation on Monitoring Multivariate Linear Profiles‖. International Journal off Industrial Engineering & Production Research, Vol. 23, No. 3, pp. 187—193
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[43] Kim, D.G. and Park, M.G. (2014), ―Horizontal Integration between cyber physical system based on Industry 4. and manufacturing execution systems through middleware building‖, Journal of Korea Multimedia Society, 17(2), pp. 1484-1493 [44] Alyahya, S., Wang, Q., Bennett, N. (2016) ―Application and integration of an RFID-enabled warehousing management system-a feasibility study.‖ Journal of Industrial Information Integration 4: 15-25.
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[45] Branger, J., Pang, Z. (2015) ―From automated home to sustainable, healthy and manufacturing home: a new story enabled by the Internet-of-Things and Industry 4.0.‖ Journal of Management Analytics 2(4): 314-332
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[46] Lee J.C., Chang T.C., and Krishnaiah P.R. (1977).‖ Approximations to the distributions of the likelihood ratio statistics for testing certain structures on the covariance matrices of real multivariate normal populations‖. Multivariate Analysis, Vol. 4, pp.105-118.
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[47] Noorossana R., Amiri A. , saghaei A. (2011). ―Statistical Analysis of Profile‖. First Edition, John Wiley: New Jersey. [48] Johnson RA, and Wichern DW.(2007). ―Applied multivariate statistical analysis‖, sixth Edition. Englewood Cliffs, NJ: Prentice hall.
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[49] Harold F., Giles J., Eldridge M. (2005) ―Extrusion: The Definitive Processing Guide and Handbook‖.
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William Andrew
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Appendix A ^
Matrix Bf :
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-0.9067 -0.9064 -0.9064 -0.9055 -0.9055 -0.9061 -0.9061 -0.9065 -0.9065 -0.9063 -0.9063 -0.9097 -0.9097 -0.9093 -0.9093 -0.9093 -0.9093 -0.911 -0.911 -0.9112 -0.9112 -0.91 -0.91 -0.9097 -0.9097 -0.9083 -0.9083 -0.9112 -0.9112 -1.8192
-0.7522 -0.7517 -0.7517 -0.7521 -0.7521 -0.7517 -0.7517 -0.7517 -0.7517 -0.7516 -0.7516 -0.7531 -0.7531 -0.7529 -0.7529 -0.7526 -0.7526 -0.7545 -0.7545 -0.7543 -0.7543 -0.755 -0.755 -0.7517 -0.7517 -0.7523 -0.7523 -0.7526 -0.7526 -1.5074
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31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
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0.0164 -0.7548 -0.7548 -0.7546 -0.7546 -0.7542 -0.7542 -0.755 -0.755 -0.7554 -0.7554 -0.7561 -0.7561 -0.7553 -0.7553 -0.7555 -0.7555 -0.7551 -0.7551 -0.7555 -0.7555 -0.7543 -0.7543 -0.7528 -0.7528 -0.7543 -0.7543 -0.7541 -0.7541 -0.7522
ED
0.0262 -0.9131 -0.9131 -0.9126 -0.9126 -0.9146 -0.9146 -0.912 -0.912 -0.9133 -0.9133 -0.9121 -0.9121 -0.9109 -0.9109 -0.9103 -0.9103 -0.9081 -0.9081 -0.908 -0.908 -0.9072 -0.9072 -0.9065 -0.9065 -0.9072 -0.9072 -0.9074 -0.9074 -0.9067
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
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Figure1. Generic components of Extruder machine (Harold [49])
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Figure 2. Dispersion diagram of profiles correlation
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0.01 0.02 0.03 0.04
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m=15
0.05
0.03 0.02
0.01
Figure4: intercept shift for sample15
0.04
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
m=5
0.01 0.02 0.03 0.04
0.03
0.05
0.02 0.01
Figure3: intercept shift for sample 5
0.04
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
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m=25
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
0.01 0.02 0.03 0.03 0.02
0.05 0.01
Figure5: intercept shift for sample25
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0.04
0.04
m=5
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m=15
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
0.02 0.03
0.04
0.04 0.02
0.02
0.03 0.03 0.02
Figure 6: Intercept shift for inverse directions of sample 5
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Figure 7: Intercept shift for inverse directions of sample 15
m=25
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0.04
0.04 0.05 0.01
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0.05 0.01
0.03
0.01
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0.01
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
0.01 0.02 0.03
0.04
0.03
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
0.03
0.05
0.02 0.01
Figure 8: Intercept shift for inverse directions of sample 25
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m=5
0.04
0.03
0.02
0.01
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
0.01 0.02 0.03 0.04
0.03
0.04
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
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0.01 0.02 0.03 0.04 0.05
probability of signal
m=15
0.05
0.02
0.01
Figure10: Slope shift for sample 15
Figure9: Slope shift for sample 5
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m=25
0.03 0.04
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0.01 0.02
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
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0.05
0.04 0.03 0.02
0.01
Figure11: Slope shift and changes in sample 25
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m=15
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m=5
0.01 0.02 0.03 0.04 0.05 0.01
0.02
0.03
0.04
Figure13: Slope shift for inverse directions of sample15
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
0.01 0.02 0.03 0.04 0.05
0.02
0.03
0.04
0.01 Figure12: Slope shift for inverse directions of sample 5
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
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probability of signal
m=25
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0.05
0.03
0.04
0.04 0.05 0.01
0.03 0.02
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0.01 0.02
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Figure14: Slope shift for inverse directions of sample 25
m=15
0.3 0.6 0.9
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1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1.2
1.2
0.9
1.5 0.3
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0.6
AC
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Figure16: standard deviation shift for sample 15
1.5
m=5
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.3 0.6 1.5
0.9
1.2
1.2
0.9 0.6
1.5 0.3
Figure15: standard deviation shift for sample 5
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m=25 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.3 0.6 1.5
0.9 1.2
0.9 1.5
0.6 0.3
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1.2
Figure17: Standard deviation shifts for sample 25
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probability of signal
m=15
0.3
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0.6 0.9 1.2
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1.5
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1.5 1.2 0.9
0.6
0.3
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Figure18: standard deviation shift for inverse directions of sample 15